Irish Math. Soc. Bulletin
Number 75, Summer 2015, 77–80
ISSN 0791-5578
Leiba Rodman: Topics in Quaternion Linear Algebra,
Princeton University Press, 2014.
ISBN:978-0-691-16185-3, USD 79.50, 384 pp.
REVIEWED BY RACHEL QUINLAN
This book consists of a comprehensive account of matrix theory
over real quaternion division algebra H. The first seven chapters
present an adaptation of many of the essential principles of matrix
theory (over the real and complex fields) to the quaternion context. Chapters 8 through 14 are mostly devoted to the theme of
quaternion matrix pencils and are closely based on a series of recent
research articles by the author. The exposition is detailed and careful, and readers familiar with concepts such as canonical forms and
standard matrix factorizations and with basic knowledge of analysis
and topology will find it accessible and largely self-contained.
Irish readers may be disappointed by the fact that an entire book
on the algebra of quaternions mentions William Rowan Hamilton
only once, in passing, as a warrant for the use of the symbol H. Otherwise the book is extraordinarily comprehensive. The first chapter
introduces some notational and other conventions, and the second
discusses the basic arithmetic of quaternions, including such topics
as automorphisms and involutions of the quaternion algebra. Particular matrix realizations of H via the real and complex regular representations are presented; these concrete interpretations of H are
often used throughout the text for computational purposes. Readers who enjoy random mathematical challenges may like Exercise
2.7.21, which asks for a demonstration that every non-zero element
of H can be written in infinitely many ways as the product of two
pure quaternions.
Chapter 3 establishes some basic principles of linear algebra for
vector spaces over the quaternions and for matrices with quaternion
entries. Standard matrix factorizations over R and C that extend
unproblematically to the quaternion setting are presented, such as
Received on 4-6-2015.
c
2015
Irish Mathematical Society
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RACHEL QUINLAN
the Gaussian elimination algorithm, the singular value decomposition, and the QR factorization for a square quaternion matrix.
A detailed analysis of topological and geometric properties of numerical ranges of quaternion matrices is provided, with respect to
general involutions as well as the standard quaternion conjugation.
Your (routine but informative) challenge from the exercises in this
chapter is to show that every element of Hn×n that has a right inverse also has a left inverse and that these coincide, and to calculate
the inverse in H3×3 of the matrix


0
i j
 −i
0 k .
−j −k 0
The next two chapters consider canonical forms, for congruence
classes of hermitian and skew-hermitian quaternion matrices (Chapter 4) and for similarity classes of square quaternion matrices (Chapter 5). A quaternion matrix A is said to be hermitian if it equal to
its conjugate transpose A∗ , and skew-hermitian if A = −A∗ . If
A ∈ Hn×n is hermitian then x∗ Ax ∈ R for all x ∈ Hn×1 , and so
concepts such as positive definiteness for hermitian matrices extend
unproblematically to the quaternion setting. Canonical forms are
established in Chapter 4 for quaternion matrices that are hermitian
or skew-hermitian, in the above sense or with respect to an involution φ other than the standard conjugation. The development and
the results appear to be analagous to the theory over C. Chapter
5 introduces the left and right spectra of a square quaternion matrix and the quaternion analogue of the Jordan canonical form. If
A ∈ Hn×n , then λ ∈ H is a right eigenvalue of A if Av = vλ for some
nonzero v ∈ Hn×1 , and µ ∈ H is a left eigenvalue of A if Au = µu
for some nonzero u ∈ Hn×1 . It is easily confirmed that if λ is a
right (or left) eigenvalue of A then so also is every element α−1 λα
of the conjugacy class of λ in the multiplicative group H× of the
quaternion division algebra. So eigenvalues of quaternion matrices
are not really single elements but conjugacy classes. There is no
general connection between the left and right spectra of an element
of Hn×n and they may be disjoint. There is a Jordan canonical
form theorem for quaternion matrices; it resembles the usual statement for algebraically closed fields except that the eigenvalues that
appear in the Jordan blocks are determined only up to conjugacy
in H× . The exercises in Chapter 5 reveal some unsettling failures
BOOK REVIEW
79
of reliable certainties of linear algebra to translate
to the division
1 i
ring setting. The harmless-looking example
shows that a
j k
quaternion matrix need not be similar to its transpose, and even
that the transpose of an invertible quaternion matrix need not be
invertible. One of the issues here seems to be that it is possible for
a set of column vectors to be right linearly independent but not left
linearly independent over H.
Chapter 6 considers subspaces of Hn×1 that are simultaneously
invariant for one matrix in Hn×n and have special properties (such
as being totally isotropic) with respect to a form defined by another.
Chapter 7 discusses the Smith normal form of a matrix written
over the ring of polynomials in one (central) variable over H, and
the Kronecker canonical form for a quaternion matrix pencil. This
theme serves as an introduction to the second half of the book, which
is concerned with canonical forms of pencils of quaternion matrices
of special forms (for example hermitian or skew-hermitian, or φ(skew)-hermitian for a nonstandard involution φ). This detailed
and extensive analysis is mostly drawn or adapted from a series of
recent research articles by the author (and collaborators in some
cases). Many open problems are included.
The book covers a huge range of material, from basic information about the algebra of quaternions to discussions of much more
specialized interest in the later chapters. It is very well organized
and the quality of exposition is high; care has been taken to provide
an accessible and readable account with plenty of interesting problems. Each chapter concludes with some notes explaining aspects of
the context and provenance of the content, and advising the reader
of relevant literature. The early chapters would certainly be useful to lecturers and students of graduate courses on such themes as
linear algebra or ring theory, and the entire volume will certainly
be useful as a reference text for researchers in linear algebra. The
author passed away in March this year, however his comprehensive
and engaging book will be appreciated long into the future.
80
RACHEL QUINLAN
Rachel Quinlan is a Senior Lecturer in Mathematics at NUI Galway. She
completed her PhD in 2000 at the University of Alberta and has research interests
in linear algebra and representation theory.
School of Mathematics, Statistics and Applied Mathematics, NUI
Galway
E-mail address: rachel.quinlan@nuigalway.ie